131 
roots. In this case three roots only of the biquadratic are 
generally equal. 
“If the cubic equation has one positive root and two 
negative roots, the biquadratic equation has two real roots 
and two imaginary roots, or else four imaginary roots.” 
(Todhunter's Equations, page 115.) 
It would seem from (19) that the four roots of the 
biquadratic are always imaginary in this case, except when 
the two negative roots of the cubic are equal ; and then the 
biquadratic has two imaginary roots and two real and equal 
roots. 
If the cubic has one positive root and two imaginary 
roots, the biquadratic has two real and two imaginary roots. 
(Tod hunter’s Equations, page 115.) This is readily proved 
by expanding y/ p + ^ 
The sum of the squares of each root of (2) is equal to the 
sum of three roots of the auxiliary cubic. 
4. The form (C) is said to be due to Simpson, Waring, 
and Ferrari, but it is certainly found in the Universal 
Arithmetic by Sir I. Newton, in Maclaurin’s Algebra, as 
well as in Simpson’s Tracts. It differs but little from the 
form (B), to which it is readily reduced as follows : — 
x^ + dx + a = rapt.x + \/ mv) 
+ (^ — \/ mjS)^ + a — \/ rnv\ mj3)^ + a + \/ - 0. . . (20) 
This reduces by multiplication to 
x^ + 2^aj® 4- (^^ + 2a-‘ m(^)x^ + 2(^a — m \/ [iy)x + — 'mv = 0 ...(21) 
An equation which coincides with (1) when the following 
conditions are satisfied : 
2d = '11^ 
+ 2a - m(3 = a 
ah-m.ypv = ±^\ ....( 22 ) 
a^-mv = c J 
These equations give 
m/3 = 2a + - a 
mv = - c 
